A quasi-static model of drop impact

Abstract

We develop a conceptually simple theoretical model of non-wetting drop impact on a rigid surface at small Weber numbers. Flat and curved impactor surfaces are considered, and the influence of surface curvature is elucidated. Particular attention is given to characterizing the contact time of the impact and the coefficient of restitution, the goal being to provide a reasonable estimate for these two parameters with the simplest model possible. Approximating the shape of the drop during impact as quasi-static allows us to derive the governing differential equation for the droplet motion from a Lagrangian. Predictions of the resulting model are shown to compare favorably with previously reported experimental results.

Exotic states of bouncing and walking droplets

“…7(d) shows the trajectory of a mode switcher settling into the high energy (2, 1) 2 mode after being perturbed by an approach to the boundary at nearly normal incidence. We note that we might alternatively have denoted the mixed state by a purely periodic mode, (24,12); however, we find it useful to distinguish between the two phases of its motion ((2, 1) 1 and (2, 1) 2 ), in which its speed is markedly different.…”

“…The vertical interaction between the bouncing drop and the liquid bath during drop contact was described using a logarithmic spring model, which built upon their model of drop impact on a rigid substrate. 24 Molacek and Bush 13 (henceforth MB2) extended their theoretical model in order to capture the dynamics of walking droplets. Specifically, their logarithmic spring model was supplemented by consideration of the wave field of the bath, which may destabilise the stationary bouncing states.…”

Pilot-Wave Hydrodynamics

“…For lowWeber-number impacts, W e = ρU 2 a 3 /σ 1, the distortion is weak, and the drop behaves roughly like a linear spring with a spring constant proportional to σ (Okumura et al 2003): during impact, kinetic energy is converted to surface energy, then back to kinetic energy, with only a small viscous loss provided R e 1. Moláček & Bush (2012) demonstrate that for the parameter regime of walkers (R e ∼ 20, B o ∼ 0.1, W e ∼ 0.1), the distortion induced by impact on a rigid substrate may be described in terms of a family of quasi-static forms, the collision dynamics in terms of a logarithmic spring.…”

“…To describe the impact of a bouncing drop on a vibrating liquid bath, Moláček & Bush (2013ab) developed a hierarchy of models of increasing complexity, building upon their model of impact on a rigid substrate (Moláček & Bush 2012) through consideration of the bath deformation. A logarithmic spring model again emerged, their model encorporating the measured logarithmic dependence of the coefficient of restitution C R and contact time T C on W e .…”

Faraday pilot-wave dynamics: modelling and computation

“…To leading order, the droplet deformation may be neglected, and the role of the interface may be treated as that of a linear spring with a spring constant proportional to the surface tension (Gilet & Bush 2010ab). Through building on a model of quasi-static droplet impact on a rigid superhydrophobic surface (Moláček & Bush 2012), Moláček & Bush (2013a) developed a model that incorporates the influence of droplet deformation and the inertia of the underlying fluid. The interface may then be described in terms of a logarithmic spring whose stiffness increases with depth of penetration.…”